Direct solve: sin²/cos² substitution

A question is this type if and only if it asks directly (no 'show' part) to solve an equation that requires using sin²θ+cos²θ=1 to convert to a quadratic in sinθ or cosθ, then solve in a given interval.

18 questions · Moderate -0.1

1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals
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CAIE P1 2022 November Q1
3 marks Moderate -0.3
1 Solve the equation \(8 \sin ^ { 2 } \theta + 6 \cos \theta + 1 = 0\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P1 2016 June Q2
4 marks Moderate -0.3
2 Solve the equation \(3 \sin ^ { 2 } \theta = 4 \cos \theta - 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2005 November Q1
4 marks Moderate -0.3
1 Solve the equation \(3 \sin ^ { 2 } \theta - 2 \cos \theta - 3 = 0\), for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2010 November Q3
4 marks Standard +0.3
3 Solve the equation \(15 \sin ^ { 2 } x = 13 + \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2012 November Q3
4 marks Moderate -0.3
3 Solve the equation \(7 \cos x + 5 = 2 \sin ^ { 2 } x\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Edexcel C2 Specimen Q4
7 marks Standard +0.3
4. Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation \(3 \sin ^ { 2 } x = 1 + \cos x\), giving your answers to the nearest degree.
OCR C2 Q4
7 marks Moderate -0.3
4. Solve the equation $$\sin ^ { 2 } \theta = 4 \cos \theta$$ for values of \(\theta\) in the interval \(0 \leq \theta \leq 360 ^ { \circ }\). Give your answers to 1 decimal place.
OCR MEI C2 Q7
5 marks Moderate -0.3
7 Showing your method clearly, solve the equation \(4 \sin ^ { 2 } \theta = 3 + \cos ^ { 2 } \theta\), for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR MEI C2 Q9
5 marks Standard +0.3
9 Showing your method, solve the equation \(2 \sin ^ { 2 } \theta = \cos \theta + 2\) for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR MEI C2 Q4
5 marks Standard +0.3
4 Showing your method clearly, solve the equation $$5 \sin ^ { 2 } \theta = 5 + \cos \theta \quad \text { for } 0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ } .$$
OCR C2 2012 June Q4
6 marks Moderate -0.3
4 Solve the equation $$4 \cos ^ { 2 } x + 7 \sin x - 7 = 0$$ giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR MEI C2 2011 January Q8
5 marks Moderate -0.3
8 Showing your method clearly, solve the equation $$5 \sin ^ { 2 } \theta = 5 + \cos \theta \quad \text { for } 0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ } .$$
OCR MEI AS Paper 1 2023 June Q4
5 marks Moderate -0.3
4 In this question you must show detailed reasoning.
Solve the equation \(6 \cos ^ { 2 } x + \sin x = 5\), giving all the roots in the interval \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
OCR MEI AS Paper 2 2024 June Q14
6 marks Standard +0.3
14 In this question you must show detailed reasoning.
Solve the equation \(5 - \cos \theta - 6 \sin ^ { 2 } \theta = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\). Turn over for question 15
Edexcel C2 Q3
8 marks Moderate -0.3
3. Find the values of \(\theta\), to 1 decimal place, in the interval \(- 180 \leq \theta < 180\) for which $$2 \sin ^ { 2 } \theta ^ { \circ } - 2 \sin \theta ^ { \circ } = \cos ^ { 2 } \theta ^ { \circ }$$ [P1 January 2002 Question 3]
Edexcel C2 Q4
7 marks Moderate -0.3
4. Solve, for \(0 \leq x < 360\), the equation $$3 \cos ^ { 2 } x ^ { \circ } + \sin ^ { 2 } x ^ { \circ } + 5 \sin x ^ { \circ } = 0$$
Edexcel C2 Q4
7 marks Moderate -0.3
4. Solve the equation $$\sin ^ { 2 } \theta = 4 \cos \theta ,$$ for values of \(\theta\) in the interval \(0 \leq \theta \leq 360 ^ { \circ }\).
AQA C4 2016 June Q2
5 marks Standard +0.3
By forming and solving a suitable quadratic equation, find the solutions of the equation $$3 \cos 2\theta - 5 \cos \theta + 2 = 0$$ in the interval \(0° < \theta < 360°\), giving your answers to the nearest \(0.1°\). [5 marks]